Multicollinearity Problem Is there a way to determine which variable has more influence on dependent variable when your variables are highly correlated between themselves ? 
Most of the ways which deals with multicollinearity helps to improve prediction, like dropping variables who have high Variance inflation factor or combining variables who have high collinearity into single predictor, but they do not answer question which variable is more important.     
 A: Assuming you're doing OLS ($Y=X\beta_{ols}+\epsilon$), here's one approach using principal component analysis (PCA) that may provide some insight:
1) Demean and divide each of your independent variables by their standard deviation - this will allow easier interpretation later on.
2) Perform PCA on your $k$ number of independent variables, with $n$ observations. This will decompose them into scores, an $n$ by $k$ matrix $S$, and a $k$ by $k$ square transformation matrix $T$:
$$
X=ST
$$ 
The scores have the property that each column is completely uncorrelated with every other column, so that $cor(s_j,s_i)=0$ for all values of $j$ and $i$, and the transformation matrix $T$ will have the property of being invertable. Furthermore, the first column of $S$, lets say $s_1$, will be the most important in terms of explaining the commonalities between columns of $X$ (this is because the first column is associated with the largest eigenvalue).
3) Estimate your model using the scores:
$$
Y=S\beta_{pca}+\epsilon
$$ 
The coefficients are not immediately interpretable, so next is where the trick comes in.
4) Look at the $j^{th}$ column of $T^{-1}$ corresponding to the $\beta_{pca,j}$ with largest t-statistic - this will tell you what relationships within $X$ are most important. The reason is made clear after re-writing identities above:
$$
\hat{Y}=X\beta_{ols}=(S)\beta_{pca}=(XT^{-1})\beta_{pca}$$so$$ \beta_{ols}=T^{-1}\beta_{pca}
$$
Because of the first step, the magnitudes of the elements in $T^{-1}$ are directly comparable, and each column denotes a linear combination of your $X$s. You're interested in what linear combinations are most relevant, so that, for example, if one variable is the most important, the $j^{th}$ column of $T^{-1}$ will have a an element in it much larger than the rest of the elements, and will correspond to the "most important" variable in $X$.
